Consider a two-person zero-sum search game between a Hider and a Searcher. The Hider chooses to hide in one of $n$ discrete locations (or "boxes") and the Searcher chooses a search sequence specifying which order to look in these boxes until finding the Hider. A search at box $i$ takes $t_i$ time units and finds the Hider - if hidden there - independently with probability $q_i$, for $i=1,\ldots,n$. The Searcher wants to minimize the expected total time needed to find the Hider, while the Hider wants to maximize it. It is shown in the literature that the Searcher has an optimal search strategy that mixes up to $n$ distinct search sequences with appropriate probabilities. This paper investigates the existence of optimal pure strategies for the Searcher - a single deterministic search sequence that achieves the optimal expected total search time regardless of where the Hider hides. We identify several cases in which the Searcher has an optimal pure strategy, and several cases in which such optimal pure strategy does not exist. An optimal pure search strategy has significant practical value because the Searcher does not need to randomize their actions and will avoid second guessing themselves if the chosen search sequence from an optimal mixed strategy does not turn out well.

翻译：考虑一个由隐藏者与搜索者构成的两人零和搜索博弈。隐藏者选择隐藏在$n$个离散位置（或"盒子"）之一，而搜索者需确定一个搜索序列——即依次检查这些盒子的顺序——直至找到隐藏者。对每个盒$i$（$i=1,\ldots,n$），每次搜索耗时$t_i$单位时间，且若隐藏者恰在该盒内，则搜索以概率$q_i$独立地发现目标。搜索者旨在最小化找到隐藏者所需的期望总时间，而隐藏者则试图最大化该时间。现有文献表明，搜索者存在一种最优混合策略，能以适当概率混合使用至多$n$种不同的搜索序列。本文研究搜索者是否存在最优纯策略——即一个确定性的搜索序列，无论隐藏者藏于何处，均能实现最优期望总搜索时间。我们识别了若干情形：当搜索者拥有最优纯策略时，以及当此类最优纯策略不存在时。最优纯搜索策略具有重要实用价值，因其免除了搜索者随机化行为的需要，且可避免因从最优混合策略中选定的搜索序列效果不佳而导致的自我怀疑。